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 A216254 Numerators of coefficients in expansion of x/((x^2+1)*arctan(x)), even powers only. 2
 1, -2, 26, -502, 7102, -44834, 295272982, -122850554, 19437784634, -83457787614326, 13505836484182762, -83261125331410322, 1230729837542663167546, -279990740971966317602, 31893076454808467404426 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The denominators are given in A225149. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..100 D. Kruchinin and V. Kruchinin, A Generating Function for the Diagonal T2n,n in Triangles, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6. FORMULA a(n) = numerator((-1)^n*sum(l=0..2*n, 2^l * (sum(k=0..l, (k!*stirling2(l,k) * stirling1(l+k,l)) / (l+k)!)) * binomial(2*n,l))). a(n) = numerator(b(n)), where b(n) = (-1)^n*(1-1/(2*n+1)-sum(i=1..n-1, b(i)*(-1)^i/(2*(n-i)+1))), b(0)=1. [Vladimir Kruchinin, Aug 29 2013] EXAMPLE x/((x^2+1)*atan(x)) = 1 - 2/3*x^2 + 26/45*x^4 - 502/945*x^6 + 7102/14175*x^8 - 44834/93555*x^10 + 295272982/638512875*x^12 - 122850554/273648375*x^14 + ... MATHEMATICA a[n_] := (-1)^n*Sum[2^l*(Sum[(k!*StirlingS2[l, k]*StirlingS1[l+k, l])/(l+k)!, {k, 0, l}])* Binomial[2*n, l], {l, 0, 2*n}]; Table[a[n] // Numerator, {n, 0, 14}] (* Jean-François Alcover, Apr 30 2013, translated from Maxima *) PROG (Maxima) a(n):=(-1)^n*sum(2^l*(sum((k!*stirling2(l, k) * stirling1(l+k, l))/(l+k)!, k, 0, l)) * binomial(2*n, l), l, 0, 2*n). (PARI) x='x+O('x^66); v=Vec(x/((x^2+1)*atan(x))); vector(#v\2, n, numerator(v[2*n-1])) \\ Joerg Arndt, Apr 29 2013 CROSSREFS Sequence in context: A285026 A137100 A228411 * A177316 A255538 A302719 Adjacent sequences:  A216251 A216252 A216253 * A216255 A216256 A216257 KEYWORD sign,frac AUTHOR Vladimir Kruchinin, Mar 15 2013 STATUS approved

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Last modified December 8 14:38 EST 2019. Contains 329865 sequences. (Running on oeis4.)